Park bridge 2018#
https://icozct.tudelft.nl/TUD_CT/CM5/collegestof/files/part4-cables-and-arches.pdf
import sympy as sp
w = sp.symbols('w', cls=sp.Function)
C1, C2 = sp.symbols('C1 C2')
x, H = sp.symbols('x H')
alpha = 1 #with axial deformation
AXexact = 1 #exact formulation of axial deformation
Ec = 30e3
Ac = 2500*280
EA = Ac*Ec/1000 #kN
L = 72
f = 1.3
q = sp.nsimplify(2.5*28*0.28) #nsimplify to get rid of floats (difficult for integration)
diffeq = sp.Eq(-H*sp.diff(w(x),x,2),q)
display(diffeq)
w = sp.dsolve(diffeq)
w = w.rhs
eq1 = sp.Eq(w.subs(x , 0) , 0)
eq2 = sp.Eq(w.subs(x , L) , 0)
sol = sp.solve((eq1 , eq2) ,
(C1 , C2))
w_sol = w.subs(sol)
display(w_sol)
dL = sp.integrate((1+sp.diff(w_sol,x)**2),(x,0,L))*alpha*H/EA
eq3 = sp.Eq(w_sol.subs(x,L/2),f)
H_sol = sp.solve(eq3,H)[0]
print("H=",H_sol,"kN")
w_sol2 = w_sol.subs(H,H_sol)
LL = sp.integrate(sp.nsimplify(sp.sqrt(1+sp.diff(w_sol2,x)**2)),(x,0,L))-dL.subs(H,H_sol) #nsimplify to get rid of floats (difficult for integration)
print("LL=",sp.N(LL),'m')
\[\displaystyle - H \frac{d^{2}}{d x^{2}} w{\left(x \right)} = \frac{98}{5}\]
\[\displaystyle - \frac{49 x^{2}}{5 H} + \frac{3528 x}{5 H}\]
H= 9769.84615384615 kN
LL= 72.0289888552996 m
Caternary model
%reset -f
import sympy as sp
w = sp.symbols('w', cls=sp.Function)
C1, C2 = sp.symbols('C1 C2')
x, H = sp.symbols('x H')
alpha = 1 # with axial deformation
AXexact = 1 #exact formulation of axial deformation
Ec = 30e3
Ac = 2500*280
EA = Ac*Ec/1000 #kN
L = 72
f = 1.3
#H = 10000
q = sp.nsimplify(32.1)
diffeq = sp.Eq(-H*sp.diff(w(x),x,2),q*sp.sqrt(1+sp.diff(w(x),x)**2))
w = sp.dsolve(diffeq) #faster than Maple
w = w.rhs
display(w)
eq1 = sp.Eq(w.subs(x , 0) , 0)
eq2 = sp.Eq(w.subs(x , L) , 0)
sol = sp.nonlinsolve((eq1 , eq2) ,
(C1 , C2)) #other solver, slower than Maple but in totality faster
print("Many complex solutions are found and for the only real solution the assumption that H is real is not taken into account:")
display(sol)
C1_sol = sol.args[0][0] #get non-complex answer from set of answers
C2_sol = sol.args[0][1]
w_sol = w.subs([(C1,C1_sol),(C2,C2_sol)])
print("The function for w is:")
display(w_sol)
dL = sp.integrate((1+sp.diff(w_sol,x)**2),(x,0,L))*alpha*H/EA
eq3 = sp.Eq(sp.integrate(sp.nsimplify(sp.sqrt(1+sp.diff(w_sol,x)**2)),(x,0,L))-dL,72.02898885)
H_sol = sp.nsolve(eq3,H,10000) #algebraic answer cannot be find, in Maple neither, numerical solver is used
display(w_sol.subs(H,H_sol))
print("H =",H_sol,"kN")
\[\displaystyle C_{1} - \frac{10 H \cosh{\left(C_{2} - \frac{321 x}{10 H} \right)}}{321}\]
Many complex solutions are found and for the only real solution the assumption that H is real is not taken into account:
\[\displaystyle \left\{\left( \frac{10 H \cosh{\left(i \left(2 n \pi + \arg{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)} \right)}}{321}, \ \left\{i \left(2 n \pi + \arg{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)}\; \middle|\; n \in \mathbb{Z}\right\}\right), \left( \frac{10 H \cosh{\left(i \left(2 n \pi + \arg{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)} \right)}}{321}, \ \left\{i \left(2 n \pi + \arg{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)}\; \middle|\; n \in \mathbb{Z}\right\}\right), \left( \frac{10 H \cosh{\left(i \left(2 n \pi + \arg{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)} \right)}}{321}, \ \left\{i \left(2 n \pi + \arg{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)}\; \middle|\; n \in \mathbb{Z}\right\}\right), \left( \frac{10 H \cosh{\left(i \left(2 n \pi + \arg{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)} \right)}}{321}, \ \left\{i \left(2 n \pi + \arg{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)}\right) + \log{\left(\left|{\sqrt{e^{\frac{11556}{5 H}}}}\right| \right)}\; \middle|\; n \in \mathbb{Z}\right\}\right), \left( \frac{10 H \cosh{\left(\log{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)} \right)}}{321}, \ \log{\left(- \sqrt{e^{\frac{11556}{5 H}}} \right)}\right), \left( \frac{10 H \cosh{\left(\log{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)} \right)}}{321}, \ \log{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)}\right)\right\}\]
The function for w is:
\[\displaystyle - \frac{10 H \cosh{\left(\log{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)} - \frac{321 x}{10 H} \right)}}{321} + \frac{10 H \cosh{\left(\log{\left(\sqrt{e^{\frac{11556}{5 H}}} \right)} \right)}}{321}\]
\[\displaystyle 447.514628963519 - 446.061124564331 \cosh{\left(0.00224184522015162 x - 0.0807064279254582 \right)}\]
H = 14318.5620985150 kN